# Function in mathematics A **function** (map, mapping, correspondence, transformation) is a binary relation between two sets that associates to each element of the first set exactly one element of the second set. A function is a process (rule) that associates each element x of a set X (the domain of the function) to a single element y of a set Y (the codomain of the function), with the possibility that the domain and codomain are the same set. Functions originally represented how a varying quantity depends on another quantity; e.g. the position of a planet is a function of time. The concept of a function was formalized at the end of the 19th century in terms of set theory. A function is a relation that is right-unique and left-serial. This relation is denoted by `f (x) = y`, where the element `x` is the input (argument) value to `f` and `y` is its output value; it is also said that `y` is the image of `x` by `f`, with `x` as the pre-image. The input \*argument, `M`, is bound by a function, `f`, by a corresponding *formal parameter*, `x`, which is its *declaration occurrence*, while all the places inside the function's body, where that parameter is used, are called its *application occurrences*. ``` ┌ function application ``` ┌ function ┐ │ to an │declaration│ │ ┌ argument │ │ │ │ f(x) = x + 2x f(5) = 5 + 2\*5 │ │ │ │ parameter │ │ │ │ │ └────┴─ application (x+2x)\[5/x] │ │ │ │ occurrences substitution │ │ │ └ parameter declaration occurrence │ └ name of the function A function, just like a relation, is uniquely represented by a set of ordered pairs, called the *graph of the function*, `f = { ∀x ∈ X. ∃y ∈ Y | (x, y) }`. `∀x ∈ X. ∃y ∈ Y. (x, y) ∈ f ∧ y = f(x)` --- # Agent Instructions: Querying This Documentation If you need additional information that is not directly available in this page, you can query the documentation dynamically by asking a question. Perform an HTTP GET request on the current page URL with the `ask` query parameter: ``` GET https://mandober.gitbook.io/math-debrief/260-function-theory/topics/function-in-mathematics.md?ask= ``` The question should be specific, self-contained, and written in natural language. The response will contain a direct answer to the question and relevant excerpts and sources from the documentation. Use this mechanism when the answer is not explicitly present in the current page, you need clarification or additional context, or you want to retrieve related documentation sections.